No, each round you can move and take an action, and possibly a second action as a bonus, if you have an ability (or spell) which allows it.
Granted, there are some characters who, after a certain level or having selected a particular option, will almost always have that bonus action available. Monk's Martial Arts. Rogue's Cunning Action. Any character that is two-weapon fighting.
Where the simplicity comes in is you don't have a huge range of options to parse when you're using that bonus action, unlike the 4E minor and swift actions. For TWF or Martial Arts, you get an extra attack. That's it. Generally there's no reason not to do it, so no time spent wavering to decide. A rogue has a few more options, but there will usually be one that's a more obvious choice for the give situation in a round. I just took out this orc so I can Dash to the next one. Is there a place to Hide? Do I want to Dodge before the dragon spews fire? Even bonus action spells will be a short list, a subset of what the character has prepared/known, and have an obvious use for a given situation. Spells, of course, also require an expenditure of resources, which is another limit on their use as bonus actions.
So, still not nearly as complex as 4E minor/swift actions, which could be part of any number of powers the class has, as well as all those other things like drinking a potion, opening a door, etc. There were so many options, players felt they must have something they could spend their minor action on, often slowing down play to figure out what that was.
Now, there's no searching through all your abilities to find the absolute best use your bonus action. No "use it or lose" anxiety. There's either a fairly obvious option, or, like KM mentions for the Champion fighter with the Defense fighting style, none at all. I'd call that much simpler.
TL;DR - the complexity isn't just about the number of actions you can take in a round; it's also about the number of options you have with those actions.